Problem: The graph of the function $y=f(x)$ is shown below. For all $x > 4$, it is true that $f(x) > 0.4$. If $f(x) = \frac{x^2}{Ax^2 + Bx + C}$, where $A,B,$ and $C$ are integers, then find $A+B+C$. [asy]
import graph; size(10.9cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-5.29,xmax=5.61,ymin=-2.42,ymax=4.34;

Label laxis; laxis.p=fontsize(10);

xaxis("$x$",xmin,xmax,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis("$y$",ymin,ymax,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); real f1(real x){return x^2/(2*x^2-2*x-12);} draw(graph(f1,xmin,-2.1),linewidth(1.2),Arrows(4)); draw(graph(f1,-1.84,2.67),linewidth(1.2),Arrows(4)); draw(graph(f1,3.24,xmax),linewidth(1.2),Arrows(4));
label("$f$",(-5.2,1),NE*lsf);

// clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
draw((-2,-2.2)--(-2,4.2),dashed);
draw((3,-2.2)--(3,4.2),dashed);
draw((-5,1/2)--(5.5,1/2),dashed);
[/asy]
Since we know that $A,B,C$ are integers, we know that the vertical asymptotes occur at the vertical lines $x = -2$ and $x = 3$. Also, since the degree of the numerator and denominator of $f$ are the same, it follows that the horizontal asymptote of $f$ occurs at the horizontal line $y = 1/A$.

We see from the graph that $1/A < 1.$  Also, we are told that for sufficiently large values of $x,$ $f(x) > 0.4,$ so
\[0.4 \le \frac{1}{A} < 1.\]As $A$ is an integer, it follows that $A = 2$.

Hence, the denominator of the function is given by $Ax^2 + Bx + C = 2(x+2)(x-3) = 2x^2 - 2x - 12$. Then, $A+B+C = 2 - 2 - 12 = \boxed{-12}$.